Boundary value problems associated with singular strongly nonlinear equations with functional terms

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type (Phi(k(t)x'(t)))' + f(t, G(x)(t))rho(t, x'(t)) = 0, on a compact interval [a, b]. These equations are quite general due to the presence of a strictly increasing homeomorphism Phi, the so-called Phi-Laplace operator, of a non-negative function k, which may vanish on a set of null measure, and moreover of a functional term G(x). We look for solutions, in a suitable weak sense, which belong to the Sobolev space W-1,W-1([a, b]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.